Does pv=0 Imply pv₁=pv₂=0 in Monic Polynomial Problems?

[rapple]

Homework Statement

Let us say that p(c) is a monic polynomial such that when applied to a particular v, we have pv=0. Let V be a finite dimensional vector space. Let V be the direct sum of k invariant subspaces. Then v = v_1+...+v_k.

When I apply pv=0 does this imply that pv_1=pv_2=0.

Homework Equations

The Attempt at a Solution

p(z)=a_0+a_1z+...a_mz^m
p(T)v=a_0+a_1(T)(v_1+v_2..v_m)+...a_m(T^m)(v_1+v_2..v_m)
We know that a_0,a_1 are not all 0s.If they are then pv = 0 for all v. So we must have that a_0 = -(the rest) or the terms must cancel out each other.
or we have (T)(v_1+v_2..v_m)=0 hence T^m(v_1+v_2..v_m)=0.

This is where I am stuck

 

[HallsofIvy]

What do you mean by "applying a polynomial to a vector"?

 

[rapple]

I mean pv=0 for a particular v